System and method for frame synchronization

ABSTRACT

A system and method for calculating and applying a metric that is calculated over a binary interval that corresponds in length to a preamble. The value of the metric reflects the likelihood that the interval is the preamble. A lower value for the metric suggests that the interval is more likely to be the preamble. In an embodiment, the metric is calculated beginning at an initial location in the bitstream, and then recalculated beginning at each of several successive locations in the bitstream. This results in a set of calculated metrics. The start of the preamble is considered likely to be the initial location of the interval that corresponds to the metric having the lowest value.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is related to prior filed U.S. Provisional ApplicationNo. 61/168,773, filed on Apr. 13, 2009 which is incorporated herein byreference in its entirety.

STATEMENT OF GOVERNMENTAL INTEREST

This invention was made with U.S. Government support under the U.S. AirForce under contract number FA8650-04-D24130006. The U.S. Government hascertain rights in the invention.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention described herein relates to communications systems, and inparticular relates to frame synchronization.

2. Description of Related Art

Free-space optical communications benefit from the advances in photonicsthat have enabled fiber-based optical systems to achieve a prominentrole in wired backbone networks. Despite this success, however, physicallayer and link layer protocols developed for fiber optics systems maynot translate successfully to free-space optical applications.

This is seen in the frame synchronization process for packetcommunications over free-space optical links. Generally, a transmittersends binary data to a receiver. The binary data is organized intoframes of N bits. The receiver sees a continuous stream of bits and mustdetermine the frame structure of the received data. In particular, thereceiver needs to determine where each frame starts. This is requiredfor a number of processes, including but not limited to forward errorcorrection. The process of determining the starting point of frames isfundamental to frame synchronization.

One mechanism by which frame synchronization may be achieved involvesthe insertion of a known preamble into each frame. A transmitter may,for example, add a known preamble that is L bits long at the beginningof each frame. If the entire frame is N bits long, then the preamblewill be followed by N−L data bits. The receiver must therefore find thepreamble in each frame in order to determine the starting point of theframe. Finding the preamble is therefore required in order to attainframe synchronization.

This poses a processing problem for packet communications over a noisychannel as is the case for wireless radio and free space opticalsystems. The receiver must identify a bit pattern that matches the knownpreamble despite the fact that the received signal contains random userdata and each bit signal may be corrupted by noise. Note that thecommunications receiver must compare the received noisy signal to athreshold in order to determine whether a 0 is present or a 1 ispresent. Such a determination may only be probabilistic in a noisyenvironment. The possibly noise-corrupted random user data furthercomplicates the preamble identification because it might match orpartially match parts of the preamble.

What is needed, therefore, is a system and method by which preambleidentification may take place with a relatively high probability ofaccuracy by minimizing errors caused by noise or random user data.

BRIEF SUMMARY OF THE INVENTION

The system and method described herein applies a metric that iscalculated in a sliding window of consecutive bits in the received data.The metric calculation examines all possible locations of the preamblewithin the data frame, each window consisting of a set of consecutivebit positions whose length is equal to that of the known preamble. Thevalue of the metric reflects the likelihood that the window is thepreamble. In certain embodiments of the invention, smaller values forthe metric suggest that the window is more likely to be the preambleThus, in such embodiments, the metric is calculated beginning at aninitial location in the bitstream, and then recalculated beginning ateach of several successive locations in the bitstream. The metriccalculations are therefore performed starting at each of severalsuccessive initial locations. This results in a set of calculatedmetrics. The start of the preamble is considered to be the initiallocation of the interval that corresponds to the metric having thelowest value. The metric can therefore be viewed as a decision metric,in that its value enables a decision as to whether the window likelyrepresents the preamble.

It should be noted, however, that transformations can be applied to agiven decision metric to produce an equally useful new decision metricfor which the decision rule might be the same (i.e., choose the windowthat generates the smallest value of the decision metric) or theopposite (i.e., choose the window that generates the largest value). Forexample, creating new metrics from old metrics by adding a constant ormultiplying by a positive scalar will not change the frame startdecisions. More generally, applying any increasing function to themetrics (e.g. logarithm) will not change the frame start decisions underthe same minimum-value decision rule. On the other hand, creating newmetrics by multiplying the original metric by −1 requires that thedecisions be made instead based on the maximum, rather than minimum,value of the new metrics. In general, applying any decreasing functionof the original (minimum-value decision rule) metrics yields equallyuseful new metrics for which the maximum-value decision rule producesthe same frame start decisions. The choice of specific transformationsto apply to a given set of metrics is a matter of convenience orpreference. For reasons of brevity and clarity, the invention will bepresented in terms embodiments of the decision metrics. Transformationsof the metrics described below represent additional embodiments of theinvention.

The system and method described herein may be applied in a frame-basedcommunications system that operates in a noisy environment. One exampleof such a system is a free-space optical system using on-off keying(OOK) modulation that is detected via an optically amplified p-i-nphotodiode receiver. For such systems, the channel is a quasi-staticfading channel with additive white Gaussian noise having the propertythat the noise variance is higher for a received 1 than for a received0. Channel fading may be due to atmospheric turbulence and may be aslowly-changing stochastic process with time constants on the order ofmilliseconds or longer. As the turbulence becomes more severe, signalfades may become deep and persistent. The synchronization process andsystem described herein presumes the prepending of the data payload witha unique word sequence or preamble that identifies the start of a frame.The preamble and packet durations are usually much smaller than thecoherence time of the channel, so that as far as frame synchronizationis concerned, the channel may be modeled as an additive white Gaussiannoise channel with signal-dependent noise variance. Should the coherencetime of the channel be comparable to or smaller than the frame duration,so that the fading causes significant amplitude distortion in thereceived signal associated with the bit stream, the distortion could beestimated and corrected prior to or as part of the frame synchronizationfunction.

BRIEF DESCRIPTION OF THE DRAWINGS/FIGURES

Further embodiments, features, and advantages of the present invention,as well as the operation of the various embodiments of the presentinvention, are described below with reference to the accompanyingdrawings.

FIG. 1 is a flowchart illustrating processing of the invention,according to an embodiment.

FIG. 2 is a block diagram illustrating the context in which theinvention may operate, according to an embodiment.

FIG. 3 is a block diagram illustrating the computing context of asoftware or firmware embodiment of the invention.

DETAILED DESCRIPTION OF THE INVENTION

A preferred embodiment of the present invention is now described withreference to the figures, where like reference numbers indicateidentical or functionally similar elements. Also in the figures, theleftmost digit of each reference number corresponds to the figure inwhich the reference number is first used. While specific configurationsand arrangements are discussed, it should be understood that this isdone for illustrative purposes only. A person skilled in the relevantart will recognize that other configurations and arrangements can beused without departing from the spirit and scope of the invention. Itwill be apparent to a person skilled in the relevant art that thisinvention can also be employed in a variety of other systems andapplications.

The description that follows presents the invention in the context of afree-space optical (FSO) packet communications system. This is not meantto limit the applicability of the invention. Embodiments of theinvention may be implemented in a variety of frame-based digitalcommunications systems where noise may be an issue.

In a FSO transceiver system, a direct detection optical receiver may beused, containing an optical pre-amplifier and p-i-n photodiodephotodetector that is AC-coupled to a trans-impedance current amplifier(TIA). The modulation format may be on-off keying (OOK) with equallylikely signals. Individual received bit decisions are made by comparingsignal levels, output by the TIA, to a threshold that may be fixed at aconstant (often set to zero for low-cost receivers) or an optimalthreshold which depends on received signal levels.

The overall processing of the invention is illustrated in FIG. 1,according to an embodiment. At 110, an initial location in the receivedbitstream is chosen. At 120, the decision metric is calculated beginningat the chosen initial location. The metric is calculated as a functionof L consecutive bits, where L is equal to the length of the preamble.In an embodiment, L may be equal to 32 bits for example.

The decision metric may take any of several forms. Generally, thedecision metric includes a first term indicative of the differencebetween the signal levels of the received detected bits and thecorresponding bit signal levels expected if the preamble were present atposition μ, and a second term indicative of the difference between thesignal levels of the received detected bits and the corresponding bitsignal levels expected for the received detected bits. The first andsecond terms are dependent on noise variances associated with theexpected signal levels.

The decision metric S(μ) is defined as follows in an embodiment of theinvention:

${S(\mu)} = {{\sum\limits_{\underset{s_{i} \neq \rho_{{\mu + i},{HD}}}{0 \leq i < L}}\left( \frac{\rho_{\mu + i} - {As}_{i}}{\sigma_{s_{i}}} \right)^{2}} - {\sum\limits_{\underset{s_{i} \neq \rho_{{\mu + i},{HD}}}{0 \leq i < L}}\left( \frac{\rho_{\mu + i} - {A\;\rho_{{\mu + i},{HD}}}}{\sigma_{\rho_{{\mu + i},{HD}}}} \right)^{2}} + {\sum\limits_{\underset{s_{i} \neq \rho_{{\mu + i},{HD}}}{0 \leq i < L}}{{\log\left( \frac{\sigma_{s_{i}}}{\sigma_{\rho_{{\mu + i},{HD}}}} \right)}^{2}.}}}$

In this expression,

L=length of the preamble,

i is a bit index,

ρ_(μ+i)=signal level (referred to herein as a soft decision value) ofreceived detected bit at position μ+i,

ρ_(μ+i, HD)=binary value (referred to herein as a hard decision value)produced by bit detector for received detected bit at position μ+i,

s_(i)=i^(th) bit of the preamble, equal to 0 or 1,

A=amplitude corresponding to the difference between the expected signallevel of a received detected bit of binary value 1 and the expectedsignal level of a received detected bit of binary value 0, and

σ=standard deviation of noise associated with a received detected bithaving binary value equal to the subscript of this variable.

As would be understood by a person of ordinary skill in the art, inother embodiments of the invention, various transformations of the abovedecision metric may be used instead, as noted above.

Returning to FIG. 1, at 130 the initial location is advanced by oneposition. At 140, a determination is made as to whether this new initiallocation is within a distance or data length of the originally choseninitial location. This data length may be equal to the size N of aframe. If so then the process continues at 120, where another decisionmetric is calculated for the L consecutive bits starting at the newinitial location that was identified at 130. In an embodiment, N may beequal to 512 bits, for example. The process iterates through 120-140,with a new decision metric generated each time. Each new decision metricis therefore calculated based on an L-long window of bits, where thewindow advances by one position after each calculation. The next metriccalculation is then based on the L bits in the new window.

If, at 140, the new initial location is now N bits away from theoriginally chosen initial location, then the process continues at 150.Here, the likely start of the preamble (and the start of the frame) isidentified as the initial location for the L consecutive bits that yieldthe minimal decision metric. The process may conclude at 160.

As noted above, in alternative embodiments, the decision metric may berestated as a function where the maximal decision metric is used toidentify the likely start of the preamble. In either case, it is anextremal value of the decision metric that is used to identify the startof the preamble.

In some embodiments, it may be appropriate to restate the above metriccalculation by applying one or more assumptions. The above decisionmetric requires knowledge of the average signal level andsignal-dependent noise variances as well as soft information (analogvalues) from the bit detector. In a practical implementation, theaverage signal level and noise variances could be estimated directlyfrom average optical power measurements for a calibrated receiver, andthe detector output could be quantized by analog-to-digital (A/D)conversion with negligible loss if the number of bits in the A/Dconverter is not small.

Typically, however, the detectors used in FSO receivers for OOKmodulation provide only one bit of precision—that is, hard-decisionestimates. Under this restriction, the frame synchronization decisionmetric can be further simplified, leading to a “weighted disagreements”version of the decision rule that is more amenable to high-speedimplementation but requires higher signal-to-noise ratio to achievereliable frame synchronization.

Specifically, when ρ_(μ+i) is replaced by its ideal hard-decisioncounterpart value Aρ_(μ+1, HD), the metric reduces to a hard-decisionform,

$\sum\limits_{\underset{s_{i} \neq \rho_{{\mu + i},{HD}}}{0 \leq i < L}}{\left\lbrack {\frac{A^{2}}{\sigma_{s_{i}}^{2}} + {\log\left( \frac{\sigma_{s_{i}}}{\sigma_{\rho_{{\mu + i},{HD}}}} \right)}^{2}} \right\rbrack.}$

Moreover, if signal-dependent noise variance is ignored, then σ isconstant and the metric becomes the number of the discrepancies betweenthe received hard-decision bits and the corresponding unique word bits:

${\sum\limits_{i = 0}^{L - 1}{s_{i} \oplus \rho_{{\mu + i},{HD}}}},$where ⊕ represents modulo 2 addition.

The decision metric may be calculated using digital logic in the form ofsoftware, firmware, or hardware, or some combination thereof. A hardwareimplementation may take the form of one or more field programmable gatearrays (FPGAs). Alternatively, a hardware implementation may take theform of one or more application specific integrated circuits (ASICs).

The term software, as used herein, refers to a computer program productincluding a computer readable medium having computer program logicstored therein to cause a computer system to perform one or morefeatures and/or combinations of features disclosed herein. A softwareembodiment is illustrated in the context of a computing system 200 inFIG. 2. System 200 may include a processor 220 and a body of memory 210that may include one or more computer readable media that may storecomputer program logic 240. Memory 210 may be implemented as a read-onlymemory (ROM) device, for example. Processor 220 and memory 210 may be incommunication using any of several technologies known to one of ordinaryskill in the art, such as a bus. Computer program logic 240 is containedin memory 210 and may be read and executed by processor 220. One or moreI/O ports and/or I/O devices, shown collectively as I/O 230, may also beconnected to processor 220 and memory 210.

Computer program logic 240 includes preamble decision logic 250. In theillustrated embodiment, preamble decision logic 250 calculates thedecision metric described above, where the metric is calculated startingat each of L successive initial locations in a received binary stream.Moreover, preamble decision logic 250 uses the calculated decisionmetric values to determine the likely beginning of a preamble in aframe. As would be known to a person of ordinary skill in the art,decision logic 250 may be implemented using any of a variety of computerprogramming languages such as, for example and without limitation, C,C++, or assembly language.

One embodiment of preamble decision logic 250 using the C programminglanguage is as follows:

min_framesync = sync_length; for (loc=0; loc<data_length; ++loc) {framesync_errors = 0; for (i=0; i<sync_length; ++i) { if(hard_decision[loc+i] != sync_pattern[i]) ++framesync_errors; } if(framesync_errors < min_framesync_errors) { min_framesync_errors =framesync_errors; estimated_framesync_loc = loc; } } // End loop overpotential frame starts return (estimated_framesync_loc);

An alternative embodiment of preamble decision logic 250 using the Cprogramming language is as follows:

min_framesync_sum = sync_length; for (loc=0; loc<data_length; ++loc) {framesync_sum = 0; for (i=0; i<sync_length; ++i) { framesync_sum +=(hard_decision[loc+i] + sync_pattern[i]%2; } if (framesync_sum <min_framesync_sum) { min_framesync_sum = framesync_sum;estimated_framesync_loc = loc; } } // End loop over potential framestarts return (estimated_framesync_loc);

The logic for implementing the decision metric calculation anddetermining the location of a preamble in a frame may be incorporated ina communications component such as a modem. This is illustrated in FIG.3, according to an embodiment. A binary signal 310 is received at amodem 320. In the illustrated embodiment, signal 310 is passed to apreamble decision module 330. At preamble decision module 330, thedecision metric described above may be calculated and the likelybeginning of the preamble, i.e., the frame boundary, may be determined.This point in the signal 310 represents the boundary of a frame. Thisinformation is shown as frame boundary 340 in the embodiment of FIG. 3.The frame boundary 340 is then used in additional frame processing ofsignal 310; logic for performing such additional frame processing isshow generically as frame processing module 350.

It is to be appreciated that the Detailed Description section, and notthe Summary and Abstract sections, is intended to be used to interpretthe claims. The Summary and Abstract sections may set forth one or morebut not all exemplary embodiments of the present invention ascontemplated by the inventor(s), and thus, are not intended to limit thepresent invention and the appended claims in any way.

The present invention has been described above with the aid offunctional building blocks illustrating the implementation of specifiedfunctions and relationships thereof. The boundaries of these functionalbuilding blocks have been arbitrarily defined herein for the convenienceof the description. Alternate boundaries can be defined so long as thespecified functions and relationships thereof are appropriatelyperformed.

While various embodiments are disclosed herein, it should be understoodthat they have been presented by way of example only, and notlimitation. It will be apparent to persons skilled in the relevant artthat various changes in form and detail may be made therein withoutdeparting from the spirit and scope of the methods and systems disclosedherein. Thus, the breadth and scope of the claims should not be limitedby any of the exemplary embodiments disclosed herein.

1. A method of performing frame synchronization, comprising: receiving a binary stream comprising a plurality of N-bit frames; and starting at an initial location μ in the binary stream, detecting a preamble of length L in the binary stream by calculating a decision metric S(μ), the decision metric computed over the bit positions within the interval μ and μ+L−1 and consisting of a plurality of terms, including a first term indicative of the difference between the signal levels of the received detected bits and the corresponding bit signal levels expected if the preamble were present at position μ, a second term indicative of the difference between the signal levels of the received detected bits and the corresponding bit signal levels expected for the received detected bits, wherein the first and second terms are dependent on noise variances associated with the expected signal levels, wherein said receiving and detecting are performed on one or more digital logic devices.
 2. The method of claim 1, wherein the decision metric is calculated in a sliding window at each of N-L consecutive bit locations and a bit location whose computed decision metric is extremal, having one of a minimum or maximum value among the N-L consecutive bit locations tested, is identified as the start of the preamble.
 3. The method of claim 2, wherein the decision metric S(μ) for starting bit location μ is calculated as a function that is an increasing function of ${M(\mu)} = {{\sum\limits_{\underset{s_{i} \neq \rho_{{\mu + i},{HD}}}{0 \leq i < L}}\left( \frac{\rho_{\mu + i} - {As}_{i}}{\sigma_{s_{i}}} \right)^{2}} - {\sum\limits_{\underset{s_{i} \neq \rho_{{\mu + i},{HD}}}{0 \leq i < L}}\left( \frac{\rho_{\mu + i} - {A\;\rho_{{\mu + i},{HD}}}}{\sigma_{\rho_{{\mu + i},{HD}}}} \right)^{2}} + {\sum\limits_{\underset{s_{i} \neq \rho_{{\mu + i},{HD}}}{0 \leq i < L}}{\log\left( \frac{\sigma_{s_{i}}}{\sigma_{\rho_{{\mu + i},{HD}}}} \right)}^{2}}}$ where is a bit index, ρ_(μ+i)=signal level of received detected bit at position μ+i, ρ_(μ+i,HD)=binary value produced by bit detector for received detected bit at position μ+i, s_(i)=ith bit of the preamble equal to 0 or 1, A=amplitude corresponding to difference between the expected signal level of a received detected bit of binary value 1 and the expected signal level of a received detected bit of binary value 0, and σ=standard deviation of noise associated with a received detected bit having a binary value equal to the subscript of this variable.
 4. The method of claim 3, wherein the initial location at which a minimal value of the decision metric is calculated is identified as the start of the preamble of a frame.
 5. The method of claim 4, wherein each soft decision value ρ_(μ+i) is approximated by the expected signal level A_(ρμ+i,HD) associated with the corresponding binary hard decision value.
 6. The method of claim 4, wherein noise variances σ² are approximated as being equal to a constant, the constant being the same whether the received detected bit is a 1 or a
 0. 7. The method of claim 2, wherein the decision metric S(μ) for starting bit location μ is calculated as a function that is an decreasing function of ${M(\mu)} = {{\sum\limits_{\underset{s_{i} \neq \rho_{{\mu + i},{HD}}}{0 \leq i < L}}\left( \frac{\rho_{\mu + i} - {As}_{i}}{\sigma_{s_{i}}} \right)^{2}} - {\sum\limits_{\underset{s_{i} \neq \rho_{{\mu + i},{HD}}}{0 \leq i < L}}\left( \frac{\rho_{\mu + i} - {A\;\rho_{{\mu + i},{HD}}}}{\sigma_{\rho_{{\mu + i},{HD}}}} \right)^{2\;}} + {\sum\limits_{\underset{s_{i} \neq \rho_{{\mu + i},{HD}}}{0 \leq i < L}}{\log\left( \frac{\sigma_{s_{i}}}{{\sigma_{\rho}}_{{\mu + i},{HD}}} \right)}^{2}}}$ where is a bit index, ρ_(μ+i)=signal level of received detected bit at position μ+i, ρ_(μ+i,HD)=binary value produced by bit detector for received detected bit at position μ+i, s_(i)=ith bit of the preamble, equal to 0 or 1, A=amplitude corresponding to difference between the expected signal level of a received detected bit of binary value 1 and the expected signal level of a received detected bit of binary value 0, σ=standard deviation of noise associated with a received detected bit having binary value equal to the subscript of this variable.
 8. The method of claim 7, wherein the initial location at which a maximal value of the decision metric is calculated is declared to be the start of the preamble of a frame.
 9. The method of claim 2, wherein the decision metric ${{S(\mu)} = {\sum\limits_{i = 0}^{L - 1}{s_{i} \oplus \rho_{{\mu + i},{HD}}}}},$ where ρ_(∥+i,HD)=binary value produced by bit detector for received detected bit at position μ+i, and s_(i)=ith bit of the preamble equal to 0 or
 1. 10. The method of claim 9, wherein the initial location at which a minimal value of the decision metric is calculated is declared to be the start of the preamble of a frame.
 11. The method of claim 2, wherein the decision metric S(μ) for starting bit location μ is calculated as a function that is a decreasing function of ${{M(\mu)} = {\sum\limits_{i = 0}^{L - 1}{s_{i} \oplus \rho_{{\mu + i},{HD}}}}},$ where ρ_(μ+i,HD)=binary value produced by bit detector for received detected bit at position μ+i, and s_(i)=ith bit of the preamble equal to 0 or
 1. 12. The method of claim 11, wherein the initial location at which a maximal value of the decision metric is calculated is declared to be the start of the preamble of the frame.
 13. A non-transitory computer readable medium including a computer readable program code having computer program logic stored thereon for causing a processor to perform frame synchronization, the computer program logic comprising: logic configured to cause the processor to start at an initial location μ in a received binary stream, and detect a preamble of length L in the binary stream by calculating a decision metric S(μ), the decision metric computed over the bit positions within the interval μ and μ+L−1 and consisting of a plurality of terms, including a first term indicative of the difference between the signal levels of the received detected bits and the corresponding bit signal levels expected if the preamble were present at position p, a second term indicative of the difference between the signal levels of the received detected bits and the corresponding bit signal levels expected for the received detected bits, wherein the first and second terms are dependent on noise variances associated with the expected signal levels.
 14. The non-transitory computer readable medium of claim 13, wherein the decision metric is calculated in a sliding window at each of N-L consecutive bit locations and a bit location whose computed decision metric is extremal, having one of a minimum or maximum value among the N-L consecutive bit locations tested, is identified as the start of the preamble.
 15. A system for performing frame processing, comprising: a preamble decision module configured to receive a binary stream comprising a plurality of N-bit frames; and starting at an initial location μ in the binary stream, detect a preamble of length L in the binary stream by calculating a decision metric S(μ), the decision metric computed over the bit positions within the interval μ and μ+L−1 and consisting of a plurality of terms, including a first term indicative of the difference between the signal levels of the received detected bits and the corresponding bit signal levels expected if the preamble were present at position a second term indicative of the difference between the signal levels of the received detected bits and the corresponding bit signal levels expected for the received detected bits, wherein the first and second terms are dependent on noise variances associated with the expected signal levels.
 16. The system of claim 15, wherein the decision metric is calculated in a sliding window at each of N-L consecutive bit locations and a bit location whose computed decision metric is extremal, having one of a minimum or maximum value among the N-L consecutive bit locations tested, is identified as the start of the preamble.
 17. The system of claim 16, wherein the decision metric S(μ) for starting bit location μ is calculated as a function that is an increasing function of ${M(\mu)} = {{\sum\limits_{\underset{s_{i} \neq \rho_{{\mu + i},{HD}}}{0 \leq i < L}}\left( \frac{\rho_{\mu + i} - {As}_{i}}{\sigma_{s_{i}}} \right)^{2}} - {\sum\limits_{\underset{s_{i} \neq \rho_{{\mu + i},{HD}}}{0 \leq i < L}}\left( \frac{\rho_{\mu + i} - {A\;\rho_{{\mu + i},{HD}}}}{\sigma_{\rho_{{\mu + i},{HD}}}} \right)^{2}} + {\sum\limits_{\underset{s_{i} \neq \rho_{{\mu + i},{HD}}}{0 \leq i < L}}{\log\left( \frac{\sigma_{s_{i}}}{\sigma_{\rho_{{\mu + i},{HD}}}} \right)}^{2}}}$ where i is a bit index, ρ_(μ+i)=signal level of received detected bit at position ρ_(μ+i,HD)=binary value produced by bit detector for received detected bit at position μ+i, s_(i)=ith bit of the preamble equal to 0 or 1, A=amplitude corresponding to difference between the expected signal level of a received detected bit of binary value 1 and the expected signal level of a received detected bit of binary value 0, σ=standard deviation of noise associated with a received detected bit having a binary value equal to the subscript of this variable.
 18. The system of claim 17, wherein the initial location at which a minimal value of the decision metric is calculated is identified as the start of the preamble of a frame.
 19. The system of claim 18, wherein each soft decision value ρ_(μ+i) is approximated by the expected signal level A_(ρμ+i,HD) associated with the corresponding binary hard decision value.
 20. The system of claim 18, wherein noise variances σ² are approximated as being equal to a constant, said constant being the same whether the received detected bit is a 1 or a
 0. 21. The system of claim 16, wherein the decision metric S(μ) for starting bit location μ is calculated as a function that is an decreasing function of ${M(\mu)} = {{\sum\limits_{\underset{s_{i} \neq \rho_{{\mu + i},{HD}}}{0 \leq i < L}}\left( \frac{\rho_{\mu + i} - {As}_{i}}{\sigma_{s_{i}}} \right)^{2}} - {\sum\limits_{\underset{s_{i} \neq \rho_{{\mu + i},{HD}}}{0 \leq i < L}}\left( \frac{\rho_{\mu + i} - {A\;\rho_{{\mu + i},{HD}}}}{\sigma_{\rho_{{\mu + i},{HD}}}\;} \right)^{2}} + {\sum\limits_{\underset{s_{i} \neq \rho_{{\mu + i},{HD}}}{0 \leq i < L}}{\log\left( \frac{\sigma_{s_{i}}}{\sigma_{\rho_{{\mu + i},{HD}}}} \right)}^{2}}}$ where is a bit index, ρ_(μ+i)=signal level of received detected bit at position μ+i, ρ_(μ+i,HD)=binary value produced by bit detector for received detected bit at position μ+i, s_(i)=ith bit of the preamble, equal to 0 or 1, A=amplitude corresponding to difference between the expected signal level of a received detected bit of binary value 1 and the expected signal level of a received detected bit of binary value 0, σ=standard deviation of noise associated with a received detected bit having binary value equal to the subscript of this variable.
 22. The system of claim 21, wherein the initial location at which a maximal value of the decision metric is calculated is identified as the start of the preamble of the frame.
 23. The system of claim 16, wherein the decision metric ${{S(\mu)} = {\sum\limits_{i = 0}^{L - 1}{s_{i} \oplus \rho_{{\mu + i},{HD}}}}},$ where ρ_(μ+i,HD)=binary value produced by bit detector for received detected bit at position μ+i, and s_(i)=ith bit of the preamble equal to 0 or
 1. 24. The system of claim 23, wherein the initial location at which a minimal value of the decision metric is calculated is identified as the start of the preamble of a frame.
 25. The system of claim 16, wherein the decision metric S(μ) for starting bit location μ is calculated as a function that is a decreasing function of ${{M(\mu)} = {\sum\limits_{i = 0}^{L - 1}{s_{i} \oplus \rho_{{\mu + i},{HD}}}}},$ where ρ_(μ+i,HD)=binary value produced by bit detector for received detected bit at position μ+i, and s_(i)=ith bit of the preamble equal to 0 or
 1. 26. The system of claim 25, wherein the initial location at which a maximal value of the decision metric is calculated is declared to be the start of the preamble of a frame. 